Pure Mathematics E4: Group Actions

29 flashcards · Mathematics · by lakesare

1. [Group Actions] How should you imagine a group action?

   Group action is turning each element of the group into a function.

2. [Group Actions] How are we: supposed to read g ∧ x VS will really read g ∧ x?

   supposed to - g wedge x ∧ is called a wedge, ∨ is called a descending wedge.will read - g acts on x

3. [Group Actions] How should we essentially think of g ∧ x?

   g(x)

4. [Group Actions] How do we denote a group action of (G, ◦) on X?

   ∧

5. What does GA stand for here?

   Group action. We call the three axioms in this definition the group action axioms.

6. [Group Actions] Name 3 properties that make ∧ a group action of (G, ◦) on X.

   GA1. Closure GA2. Identity GA3. Composition

7. [Group Actions] Visually imagine the GA1 property.

8. [Group Actions] Visually imagine the GA2 property.

9. [Group Actions] Rephrase the GA3 property.

   ◦ should behave like function composition. This is just the definition of function composition.

10. [Group Actions] When is ∧ a faithful group action?

    When each g forms a different function. Basically ∧ turns each element g of the group into a function on X. Example of the group action that is NOT faithful. Two or more elements of the group permute the elements of the set in the same way.The action of the group S(□) on the set { R, S, T, U }.

11. [Group Actions] What are g, h, etc. like 𝘢𝘴 𝘧𝘶𝘯𝘤𝘵𝘪𝘰𝘯𝘴?

    g, h, etc. are 1-1 and onto!

12. [Group Actions] What does Orb(x) stand for?

    Orbit of x

13. [Group Actions] Visually imagine Orb(x) as H as an orbit.

    The orbit of an element x

14. [Group Actions] What does Stab(x) stand for?

    Stabiliser of x

15. [Group Actions] Visually imagine Stab(x).

    The stabiliser of an element x. The stabiliser Stab x of a set element x under the action of a group always contains the identity element e of the group. This is because e ∧ x = x, by axiom GA2.

16. [Group Actions] What group actions is generally about?

    It's about making elements of the group act as functions.

17. [Group Actions] Visually imagine Orb(x) and Stab(x).

    The orbit of an element x. The stabiliser of an element x.

18. [Group Actions] How are these similar: orbits conjugacy classes right cosets?

    All of these form a partition of a set.

19. What does OS Theorem stand for?

    Orbit-Stabiliser Theorem.

20. Visually imagine the Orbit-Stabiliser Theorem.

    It's similar to Ker * Im theorem in linear algebra and in E3. The stabiliser of an element x. The orbit of an element x.

21. [The Counting Theorem] Rephrase the Bangle Problem.

22. [The Counting Theorem] Rephrase the Chessboard Problem.

23. [The Counting Theorem] Rephrase the Cube Problem.

    A cube (see Figure 60) has six faces, and in this problem each of them is to be coloured with one of three colours, so by the Multiplication Principle there are 36 coloured cubes in fixed positions. However, we would want to regard two coloured cubes as the same when one can be rotated to give the other.

24. [Fixed Sets] Visually imagine Fix(g). Fixed set of g.

    The fixed set of a group element g. The fixed point sets that you met in Subsection 4.1 of Unit E2 are special cases of fixed sets. You saw there that if f is a symmetry of a figure F, then the fixed point set of f is the set of all points of F that are fixed by f. This is the fixed set of f under the natural action of the symmetry group S(F ) on the set of points in F.

25. Visually imagine Fix(g) VS Stab(x). In particular, Fix g is a subset of X, whereas Stab x is a subgroup of G.

    Fix(g) - fixed set of g. The fixed set of a group element gStab(x) - stabiliser of x. The stabiliser of an element x.

27. [Group Actions] TODO

28. [Group Actions] What does GA stand for?

    Group Action GA1/2/3 are properties of (G, ⚬) and X for ∧ to be a group action. Group action is turning each element of the group into a function.

29. Goddamn example of a group action TODO